Mountain biker While mountain biking, you first move at constant speed along the bottom of a trail’s circular dip and then at constant speed across the top of a circular hump. Assume that you and the bike are a system. Determine the direction of the acceleration at each position and construct a force diagram for each position (consistent with the direction of the acceleration). Compare at each position the magnitude of the force of the surface on the bike with the force Earth exerts on the system

Andy C.

Numerade Educator

You swing a rock tied to a string in a vertical circle. (a) Determine the direction of the acceleration of the rock as it passes the lowest point in its swing. Construct a consistent force diagram for the rock as it passes that point. How does the force that the string exerts on the rock compare to the force that Earth exerts on the rock? Explain. (b) Repeat the above analysis as best you can for the rock as it passes the highest point in the swing. (c) If the string is tied around your finger, when do you feel a stronger pull when the rock is at the bottom of the swing or at the top? Explain

Andy C.

Numerade Educator

Loop-the-loop You ride a roller coaster with a loop-the-loop. Compare as best you can the normal force that the seat exerts on you to the force that Earth exerts on you when you are passing the bottom of the loop and the top of the loop. Justify your answers by determining the direction of acceleration and constructing a force diagram for each position. Make your answers consistent with Newton’s second law.

Andy C.

Numerade Educator

You start an old record player and notice a bug on the surface close to the edge of the record. The record has a diameter of 12 inches and completes 33 revolutions each minute. (a) What are the speed and the acceleration of the bug? (b) What would the bug’s speed and acceleration be if it were halfway between the center and the edge of the record?

Andy C.

Numerade Educator

Determine the acceleration of Earth due to its motion around the Sun. What do you need to assume about Earth to make the calculation? How does this acceleration compare to the acceleration of free fall on Earth?

Andy C.

Numerade Educator

The Moon is an average distance of $3.8 \times 10^{8} \mathrm{m}$ from Earth. It circles Earth once each 27.3 days. (a) What is its average speed? (b) What is its acceleration? (c) How does this acceleration compare to the acceleration of free fall on Earth?

Andy C.

Numerade Educator

Aborted plane landing You are on an airplane that is landing. The plane in front of your plane blows a tire. The pilot of your plane is advised to abort the landing, so he pulls up, moving in a semicircular upward-bending path. The path has a radius of 500 $\mathrm{m}$ with a radial acceleration of 17 $\mathrm{m} / \mathrm{s}^{2} .$ What is the plane's speed?

Andy C.

Numerade Educator

UItracentrifuge You are working in a biology lab and learning to use a new ultracentrifuge for blood tests. The specifications for the centrifuge say that a red blood cell rotating in the ultracentrifuge moves at 470 $\mathrm{m} / \mathrm{s}$ and has a radial acceleration of $150,000 ~ g^{\prime}$ s (that is, $150,000$ times 9.8 $\mathrm{m} / \mathrm{s}^{2} )$ The radius of the centrifuge is 0.15 $\mathrm{m} .$ You wonder if this claim is correct. Support your answer with a calculation.

Andy C.

Numerade Educator

Jupiter rotates once about its axis in 9 $\mathrm{h} 56 \mathrm{min}$ . Its radius is $7.13 \times 10^{4} \mathrm{km}$ . Imagine that you could somehow stand on the surface (although in reality that would not be possible, because Jupiter has no solid surface). Calculate your radial acceleration in meters per second squared and in Earth g's.

Andy C.

Numerade Educator

Imagine that you are standing on a horizontal rotating platform in an amusement park (like the platform for a merry-go-round). The period of rotation and the radius of the platform are given, and you know your mass. Make a list of the physical quantities you could determine using this information, and describe how you would determine them.

Andy C.

Numerade Educator

A car moves along a straight line to the right. Two friends standing on the sidewalk are arguing about the motion of a point on the rotating car tire at the instant it reaches the lowest point (touching the road). Jake says that the point is at rest. Morgan says that the point is moving to the left at the car’s speed. Justify each friend’s opinion. Explain whether it is possible for them to simultaneously be correct.

Andy C.

Numerade Educator

Three people are standing on a horizontally rotating plat-form in an amusement park. One person is almost at the edge, the second one is $(3 / 5) R$ from the center, and the third is $(1 / 2) R$ from the center. Compare their periods of rotation, their speeds, and their radial accelerations.

Andy C.

Numerade Educator

Consider the scenario described in Problem 12. If the platform speeds up, who is more likely to have trouble staying on the platform? Support your answer with a force diagram and describe the assumptions that you made.

Andy C.

Numerade Educator

Merry-go-round acceleration Imagine that you are standing on the rotating platform of a merry-go-round in an amusement park. You have a stopwatch and a measuring tape. Describe how you will determine your radial acceleration when standing at the edge of the platform and when halfway from the edge. What do you expect the ratio of these two accelerations to be?

Andy C.

Numerade Educator

Ferris wheel You are sitting on a rotating Ferris wheel. Draw a force diagram for yourself when you are at the bottom of the circle and when you are at the top.

Andy C.

Numerade Educator

Estimate the radial acceleration of the foot of a college football player in the middle of punting a football.

Andy C.

Numerade Educator

Estimate the radial acceleration of the toe at the end of the horizontally extended leg of a ballerina doing a pirouette.

Andy C.

Numerade Educator

Is it safe to drive your 1600-kg car at speed 27 m/s around a level highway curve of radius 150 m if the effective coefficient of static friction between the car and the road is 0.40?

Andy C.

Numerade Educator

You are fixing a broken rotary lawn mower. The blades on the mower turn 50 times per second. What is the magnitude of the force needed to hold the outer 2 cm of the blade to the inner portion of the blade? The outer part is 21 cm from the center of the blade, and the mass of the outer portion is 7.0 g.

Andy C.

Numerade Educator

Your car speeds around the 80-m-radius curved exit ramp of a freeway. A 70-kg student holds the armrest of the car door, exerting a 220-N force on it in order to prevent himself from sliding across the vinyl-covered back seat of the car and slamming into his friend. How fast is the car moving in meters per second and miles per hour? What assumptions did you make?

Andy C.

Numerade Educator

How fast do you need to swing a 200 -g ball at the end of a string in a horizontal circle of 0.5 -m radius so that the string makes a $34^{\circ}$ angle relative to the horizontal? What assumptions did you make?

Andy C.

Numerade Educator

Christine’s bathroom scale in Maine reads 110 lb when she stands on it. Will the scale read more or less in Singapore if her mass stays the same? To answer the question, (a) draw a force diagram for Christine. (b) With the assistance of this diagram, write an expression using Newton’s second law that relates the forces exerted on Christine and her acceleration along the radial direction. (c) Decide whether the reading of the scale is different in Singapore. List the assumptions that you made and describe how your answer might change if the assumptions are not valid.

Andy C.

Numerade Educator

A child is on a swing that moves in the horizontal circle of radius 2.0 m depicted in Figure P4.23. The mass of the child and the seat together is 30 kg and the two cables exert equal-magnitude forces on the chair. Make a list of the physical quantities you can determine using the sketch and the known information. Determine one kinematics and two dynamics quantities from that list.

Andy C.

Numerade Educator

A coin rests on a record 0.15 m from its center. The record turns on a turntable that rotates at variable speed. The coefficient of static friction between the coin and the record is 0.30. What is the maximum

coin speed at which it does not slip?

Shreya N.

Numerade Educator

Roller coaster ride A roller coaster car travels at speed 8.0 $\mathrm{m} / \mathrm{s}$ over a 12 -m-radius vertical circular hump. What is the magnitude of the upward force that the coaster seat exerts on a 48 -kg woman passenger?

Andy C.

Numerade Educator

A person sitting in a chair (combined mass 80 $\mathrm{kg} )$ is attached to a 6.0 -m-long cable. The person moves in a horizontal circle. The cable angle $\theta$ is $62^{\circ}$ below the horizontal. What is the person's speed? Note: The radius of the circle is not 6.0 $\mathrm{m}$ .

Andy C.

Numerade Educator

A car moves around a 50 -m-radius highway curve. The road, banked at $10^{\circ}$ relative to the horizontal, is wet and icy so that the coefficient of friction is approximately zero. At what speed

should the car travel so that it makes the turn without slipping?

Andy C.

Numerade Educator

A 20.0 -g ball is attached to a $120-\mathrm{cm}$ -long string and moves in a horizontal circle (see Figure $P 4.28 ) .$ The string exerts a force on the ball that is equal to 0.200 $\mathrm{N} .$ What is

the angle $\theta ?$

Andy C.

Numerade Educator

A 50-kg ice skater goes around a circle of radius 5.0 m at a constant speed of 3.0 m/s on a level ice rink. What are the magnitude and direction of the horizontal force that the ice exerts on the skates?

Andy C.

Numerade Educator

A car traveling at 10 m/s passes over a hill on a road that has a circular cross section of radius 30 m. What is the force exerted by the seat of the car on a 60-kg passenger when the car is passing the top of the hill?

Andy C.

Numerade Educator

A 1000-kg car is moving at 30 m/s around a horizontal level curved road whose radius is 100 m. What is the magnitude of the frictional force required to keep the car from sliding?

Zulfiqar A.

Numerade Educator

Equation Jeopardy 1 Describe using words, a sketch, a velocity change diagram, and a force diagram two situations whose mathematical description is presented below.

$$

700 \mathrm{N}-(30 \mathrm{kg})\left(9.8 \mathrm{m} / \mathrm{s}^{2}\right)=\frac{(30 \mathrm{kg}) v^{2}}{12 \mathrm{m}}

$$

Zulfiqar A.

Numerade Educator

Equation Jeopardy 2 Describe using words, a sketch, a velocity change diagram, and a force diagram two situations whose mathematical description is presented below.

$$

\frac{(2.0 \mathrm{kg})\left(4.0 \mathrm{m}^{2} / \mathrm{s}^{2}\right)}{r}=0.4 \times(2.0 \mathrm{kg}) \times(9.8 \mathrm{N} / \mathrm{kg})

$$

Zulfiqar A.

Numerade Educator

Banked curve raceway design You need to design a banked curve at the new circular Super 100 Raceway. The radius of the track is 800 m and cars typically travel at speed 160 mi/h. What feature of the design is important so that all race cars can move around the track safely in any weather? (a) Provide a quantitative answer. (b) List your assumptions and describe whether the number you provided will increase or decrease if the assumption is not valid

Zulfiqar A.

Numerade Educator

A circular track is in a horizontal plane, has a radius of $r$ meters, and is banked at an angle $\theta$ above the horizontal. (a) Develop an expression for the speed a person should rollerblade on this track so that she needs zero friction to prevent her from sliding sideways off the track. (b) Should another person move faster or slower if her mass is 1.3 times the mass of the first person? Justify your answer.

Zulfiqar A.

Numerade Educator

Design a quantitative test for Newton’s second law as applied to constant speed circular motion. Describe the experiment and provide the analysis needed to make a prediction using the law

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pin-dry cycle Explain how the spin-dry cycle in a washing machine removes water from clothes. Be specific.

Zulfiqar A.

Numerade Educator

Your friend says that the force that the Sun exerts on Earth is much larger than the force that Earth exerts on the Sun. (a) Do you agree or disagree with this opinion? (b) If you disagree, how would you convince him of your opinion?

Zulfiqar A.

Numerade Educator

Determine the gravitational force that (a) the Sun exerts on the Moon, (b) Earth exerts on the Moon, and (c) the Moon exerts on Earth. List at least two assumptions for each force that you made when you calculated the answers.

Zulfiqar A.

Numerade Educator

(a) What is the ratio of the gravitational force that Earth exerts on the Sun in the winter and the force that it exerts in the summer? (b) What does it tell you about the speed of Earth during different seasons? (c) How many correct answers can you give for part (a)? Hint: Earth’s orbit is an ellipse with the Sun located at one of the foci of the ellipse.

Zulfiqar A.

Numerade Educator

Black hole gravitational force A black hole exerts a $50-\mathrm{N}$ gravitational force on a spaceship. The black hole is $10^{14} \mathrm{m}$ from the ship. What is the magnitude of the force that the black hole exerts on the ship when the ship is one-half that distance from the black hole? [Hint: One-half of $10^{14} \mathrm{m}$ is not $10^{7} \mathrm{m} . ]$

Zulfiqar A.

Numerade Educator

The average radius of Earth's orbit around the Sun is $1.5 \times 10^{8} \mathrm{km} .$ The mass of Earth is $5.97 \times 10^{24} \mathrm{kg},$ and it makes one orbit in approximately 365 days. (a) What is Earth's speed relative to the Sun? (b) Estimate the Sun's mass using Newton's law of universal gravitation and Newton's second law. What assumptions did you need to make?

Zulfiqar A.

Numerade Educator

The Moon travels in a $3.8 \times 10^{5} \mathrm{km}$ radius orbit about Earth. Earth's mass is $5.97 \times 10^{24} \mathrm{kg} .$ Determine the period $T$ for one Moon orbit about Earth using Newton's law of universal gravitation and Newton's second law. What assumptions did you make?

Zulfiqar A.

Numerade Educator

Determine the ratio of Earth's gravitational force exerted on an 80 -kg person when at Earth's surface and when 1000 $\mathrm{km}$ above Earth's surface. The radius of Earth is 6370 $\mathrm{km} .$

Zulfiqar A.

Numerade Educator

Determine the magnitude of the gravitational force Mars would exert on your body if you were on the surface of Mars.

Zulfiqar A.

Numerade Educator

When you stand on a bathroom scale here on Earth, it reads 540 N. (a) What would your mass be on Mars, Venus, and Saturn? (b) What is the magnitude of the gravitational force each planet would exert on you if you stood on their surface? (c) What assumptions did you make?

Zulfiqar A.

Numerade Educator

The free-fall acceleration on the surface of Jupiter, the most massive planet, is 24.79 $\mathrm{m} / \mathrm{s}^{2} .$ Jupiter's radius is $7.0 \times 10^{4} \mathrm{km} .$ Use Newtonian ideas to determine Jupiter's mass.

Zulfiqar A.

Numerade Educator

A satellite moves in a circular orbit a distance of $1.6 \times 10^{5} \mathrm{m}$ above Earth's surface. Determine the speed of the satellite.

Zulfiqar A.

Numerade Educator

Mars has a mass of $6.42 \times 10^{23} \mathrm{kg}$ and a radius of $3.40 \times 10^{6} \mathrm{m} .$ Assume a person is standing on a bathroom scale on the surface of Mars. Over what time interval would Mars have to complete one rotation on its axis to make the bathroom scale have a zero reading?

Zulfiqar A.

Numerade Educator

Determine the speed a projectile must reach in order to become an Earth satellite. What assumptions did you make?

Zulfiqar A.

Numerade Educator

Determine the distance above Earth’s surface to a satellite that completes two orbits per day. What assumptions did you make?

Zulfiqar A.

Numerade Educator

Determine the period of an Earth satellite that moves in a circular orbit just above Earth’s surface. What assumptions did you need to make?

Zulfiqar A.

Numerade Educator

A spaceship in outer space has a doughnut shape with 500-m outer radius. The inhabitants stand with their heads toward the center and their feet on an outside rim (see Figure Q4.21). Over what time interval would the spaceship have to complete one rotation on its axis to make a bathroom scale have the same reading for the person in space as when on Earth’s surface?

Dading C.

Numerade Educator

Loop-the-loop You have to design a loop-the-loop for a new amusement park so that when each car passes the top of the loop inverted (upside-down), each seat exerts a force against a passenger’s bottom that has a magnitude equal to 1.5 times the gravitational force that Earth exerts on the passenger. Choose some reasonable physical quantities so these conditions are met. Show that the loop-the-loop will work equally well for passengers of any mass

Zulfiqar A.

Numerade Educator

A Tarzan swing Tarzan (mass 80 kg) swings at the end of an 8.0-m-long vine (Figure P4.55). When directly under the vine’s support, he releases the vine and flies across a swamp. When he releases the vine, he is 5.0 m above the swamp and 10.0 m horizontally from the other side. Determine the force

the vine exerts on him at the instant before he lets go (the vine is straight down when he lets go).

Zulfiqar A.

Numerade Educator

(a) If the masses of Earth and the Moon were both doubled, by how much would the radius of the Moon’s orbit about Earth have to change if its speed did not change? (b) By how much would its speed have to change if its radius did not change? Justify each answer.

Zulfiqar A.

Numerade Educator

Estimate the radial acceleration of the tread of a car tire. Indicate any assumptions that you made.

Zulfiqar A.

Numerade Educator

Estimate the force exerted by the tire on a 10-cmlong section of the tread of a tire as the car travels at speed 80 km/h. Justify any numbers used in your estimate

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Estimate the maximum radial force that a football player’s leg needs to exert on his foot when swinging the leg to punt the ball. Justify any numbers that you use.

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Design 1 Design and solve a circular motion problem for a roller coaster

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Design 2 Design and solve a circular motion problem for the amusement park ride shown in Figure P4.61.

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Demolition An old building is being demolished by swinging a heavy metal ball from a crane. Suppose that such a 100-kg ball swings from a 20-m-long wire at speed 16 m/s as the wire passes the vertical orientation. (a) What tension force must the wire be able to withstand in order not to break? (b) Assume the ball stops after sinking 1.5 m into the wall. What was the average force that the ball exerted on the wall? Indicate any assumptions you made for each part of the problem.

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Designing a banked roadway You need to design a banked curve for a highway in which cars make a $90^{\circ}$ turn moving at 50 $\mathrm{mi} / \mathrm{h}$ . Indicate any assumptions you make.

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Evaluation question You find the following report about blackouts. The acceleration that causes blackouts in fighter pilots is called the maximum g-force. Fighter pilots experience this force when accelerating or decelerating quickly. At high g’s, the pilot’s blood pressure changes and the flow of oxygen to the brain rapidly decreases. This happens because the pressure outside of the pilot’s body is so much greater than the pressure a human is normally accustomed to. Indicate any incorrect physics (including the application of physics to biology) that you find.

Zulfiqar A.

Numerade Educator

Suppose that Earth rotated much faster on its axis—so fast that people were almost weightless when at Earth’s surface. How long would the length of a day be on this new Earth?

Zulfiqar A.

Numerade Educator

On Earth, an average person’s vertical jump is 0.40 m. What is it on the Moon? Explain

Zulfiqar A.

Numerade Educator

You read in a science magazine that on the Moon, the speed of a shell leaving the barrel of a modern tank is enough to put the shell in a circular orbit above the surface of the Moon (there is no atmosphere to slow the shell). What should be the speed for this to happen? Is this number reasonable?

Zulfiqar A.

Numerade Educator

Why did drivers get dizzy and disoriented while driving at the Texas Motor Speedway?

(a) The cars were traveling at over 200 mi/h.

(b) The track was tilted at an unusually steep angle.

(c) On turns the drivers’ blood tended to drain from their brains into veins in their lower bodies.

(d) The g force pushed blood into their heads.

(e) The combination of a and b caused c.

Zulfiqar A.

Numerade Educator

What was the time interval needed for Gil de Ferran's car to complete one lap during his record-setting drive?

$$\begin{array}{llll}{\text { (a) } 16.8 \mathrm{s}} & {\text { (b) } 18.4 \mathrm{s}} & {\text { (c) } 22.4 \mathrm{s}}\end{array}$$

$$(d) 25.1 \mathrm{s} \quad (e) 37.3 \mathrm{s}$$

Zulfiqar A.

Numerade Educator

If the racecars had no help from friction, which expression below would describe the normal force of the track on the cars while traversing the $24^{\circ}$ banked curves?

$$

\begin{array}{ll}{\text { (a) } m g \cos 66^{\circ}} & {\text { (b) } m g \sin 66^{\circ}} & {\text { (c) } m g / \cos 66^{\circ}} \\ {\text { (d) } m g / \sin 66^{\circ}} & {\text { (e) None of these }}\end{array}

$$

Zulfiqar A.

Numerade Educator

For the racecars to stay on the road while traveling at high speed, how did a friction force need to be exerted?

(a) Parallel to the roadway and outward

(b) Parallel to the roadway and toward the infield

(c) Horizontally toward the center of the track

(d) Opposite the direction of motion

(e) None of the above

Zulfiqar A.

Numerade Educator

The average speed reported in the reading passage has six significant digits, implying that the speed is known to within $\pm 0.001 \mathrm{mi} / \mathrm{h}$ . If this is correct, which answer below is closest to the uncertainty in the time needed to travel around the 1.5 -mi oval track? (Think about the percent uncertainties.)

$$\begin{array}{ll}{\text { (a) } \pm 0.0001 \mathrm{s}} & {\text { (b) } \pm 0.001 \mathrm{s}} & {\text { (c) } \pm 0.01 \mathrm{s}}\end{array}$$

$$(d) \pm 0.1 \mathrm{s} \quad(\mathrm{e}) \pm 1 \mathrm{s}$$

Zulfiqar A.

Numerade Educator

What was the approximate radius of the part of the track where the drivers experienced the 5.5 g acceleration?

$$\begin{array}{llll}{\text { (a) } 40 \mathrm{m}} & {\text { (b) } 80 \mathrm{m}} & {\text { (c) } 200 \mathrm{m}}\end{array}$$

$$(d) 400 \mathrm{m} \quad (e) 1000 \mathrm{m}$$

Zulfiqar A.

Numerade Educator

Use the velocity change method to estimate the comet’s direction of acceleration when passing closest to the Sun (position I Figure P4.74).

(a) A

(b) B

(c) C

(d) D

(e) The acceleration is zero.

Zulfiqar A.

Numerade Educator

What object or objects exert forces on the comet as it passes position I (shown Figure P4.74)?

(a) The Sun’s gravitational force toward the Sun

(b) The force of motion tangent to the direction the comet is traveling

(c) An outward force away from the Sun

(d) a and b

(e) a, b, and c

Zulfiqar A.

Numerade Educator

Suppose that instead of being peanut shaped, Halley's Comet was spherical with a radius of 5.0 $\mathrm{km}$ (about its present volume). Which answer below would be closest to your radial acceleration if you were standing on the equator of the rotating comet?

$$\begin{array}{llll}{\text { (a) } 10^{-5} \mathrm{m} / \mathrm{s}^{2}} & {\text { (b) } 10^{-3} \mathrm{m} / \mathrm{s}^{2}} & {\text { (c) } 0.1 \mathrm{m} / \mathrm{s}^{2}}\end{array}$$

$$(d) 10 \mathrm{m} / \mathrm{s}^{2} \quad (e) 1000 \mathrm{m} / \mathrm{s}^{2}$$

Zulfiqar A.

Numerade Educator

Approximately what gravitational force would the spherical- shaped $5-\mathrm{km}$ radius comet exert on a 100 -kg person on the surface of the comet?

$$\begin{array}{ll}{\text { (a) } 0.06 \mathrm{N}} & {\text { (b) } 0.6 \mathrm{N}} & {\text { (c) } 6 \mathrm{N}}\end{array}$$

$$(d) 60 \mathrm{N} \quad (e) 600 \mathrm{N}$$

Zulfiqar A.

Numerade Educator

The closest distance that the comet passes relative to the Sun is $8.77 \times 10^{10} \mathrm{m}(\text { position I in Figure } \mathrm{P} 4.74) .$ Apply Newton's second law and the law of universal gravitation to determine which answer below is closest to the comet's speed when passing position I.

$$\begin{array}{llll}{\text { (a) } 1000 \mathrm{m} / \mathrm{s}} & {\text { (b) } 8000 \mathrm{m} / \mathrm{s}} & {\text { (c) } 20,000 \mathrm{m} / \mathrm{s}}\end{array}$$

$$(d) 40,000 \mathrm{m} / \mathrm{s} \quad (e) 800,000 \mathrm{m} / \mathrm{s}$$

Zulfiqar A.

Numerade Educator

The farthest distance that the comet is from the Sun is $5.25 \times 10^{12} \mathrm{m}(\text { position II in Figure } \mathrm{P} 4.74) .$ Apply Newton's second law and the law of universal gravitation to determine which answer below is closest to the comet's speed when passing position II.

$$\begin{array}{llll}{\text { (a) } 800 \mathrm{m} / \mathrm{s}} & {\text { (b) } 5000 \mathrm{m} / \mathrm{s}} & {\text { (c) } 10,000 \mathrm{m} / \mathrm{s}}\end{array}$$

$$(d) 50,000 \mathrm{m} / \mathrm{s} \quad (e) 80,000 \mathrm{m} / \mathrm{s}$$

Zulfiqar A.

Numerade Educator